Iterative calculation values in Collatz conjecture eventually reach 1 as if walking across tightropes: Proof of the correctness of the Collatz conjecture using the Collatz Rope Model

Abstract

I have previously reported the results of my investigations into the Collatz conjecture in two papers. In the first paper (Shiraishi, 2026a), I examined the properties of the values generated through iterative calculation, based on extensive numerical experiments. I also proposed the Collatz Rope Model and presented theoretical discussions concerning the conjecture. In the second paper (Shiraishi, 2026b), I supplemented the descriptions given in the first paper and provided additional theoretical reinforcement. However, certain descriptive inaccuracies and theoretical ambiguities remain in these papers, and I am concerned that they may hinder a clear understanding of my conclusion that the Collatz conjecture holds for all natural numbers. Therefore, the purpose of this paper is to reorganize these theoretical explanations in a concise and accessible manner, thereby demonstrating the correctness of the Collatz conjecture with greater clarity.